9.6 Spectral collocation differentiation

The equivalence (9.80) between the discrete truncated expansion and interpolation at Gauss-type points
allows the approximation of the derivative of a function in a very simple way,

Therefore, knowing the values of the function at the collocation points, i.e., the Gauss-type points, we
can construct its interpolant , take an exact derivative thereof, and evaluate the result at the
collocation points to obtain the values of the discrete derivative of at these points. This leads to a
matrix-vector multiplication, where the corresponding matrix elements can be computed once and for
all:

with

We give the explicit expressions for this differentiation matrix for Chebyshev polynomials both at Gauss
and Gauss–Lobatto points (see, for example, [167, 237]).

Chebyshev–Gauss.

with a prime denoting differentiation.

Chebyshev–Gauss–Lobatto.

where for and for .

"Continuum and Discrete Initial-Boundary Value Problems
and Einstein’s Field Equations"